In the opening set neither man offered a break point until the fifth game when Baghdatis broke at the second attempt. The Cypriot could not consolidate the advantage, though, as Wawrinka broke straight back.
The ninth seed then snapped Baghdatis's serve again at the next opportunity by finally converting his seventh break point of the eighth game to go 5-3 ahead.
Wawrinka served out the set in the next game and then broke Baghdatis in the first game of the second.
The Swiss saved break-back points at 4-2 before opening up a double-break advantage and taking the set 6-2.
In the third set there was not a single break point until the final game before the tie-break. Wawrinka saved it but in the tie-break it was Baghdatis who raced to a 7-1 success.
The Cypriot continued his momentum by breaking Wawrinka in the first game of the fourth but relinquished the advantage as Wawrinka pulled it back to 3-3.
Another tie-break was needed and this time it was Wawrinka who battled through, taking it 9-7 to reach the last 16.
Wawrinka will face fifth seed Tomas Berdych, who breezed past Julien Benneteau, in the fourth round at Flushing Meadows.